A note on equality on plurifinely open sets of some complex Monge-Ampère measures

Our aim in this paper is to prove that if plurisubharmonic functions $u_1,\mathrm{.}\mathrm{.}\mathrm{.},u_n$, $v_1,\mathrm{.}\mathrm{.}\mathrm{.},v_n$ in the domain of definition of the complex Monge-Ampère operator on a domain set $D\subset {\mathbb{C}}^n$ ($n\ge 1$) are such that $u_1=v_1,\mathrm{.}\mathrm{.}\mathrm{.},u_n=v_n$ on a Borel plurifinely open set $\mathrm{\Omega }\subset D$, then $$d{d}^c{u}_1\wedge \mathrm{.}\mathrm{.}\mathrm{.}\wedge d{d}^c{u}_n=d{d}^c{v}_1\wedge \mathrm{.}\mathrm{.}\mathrm{.}\wedge d{d}^c{v}_n$$ on $\mathrm{\Omega }$. This extends an earlier result obtained by the author in \cite{EK}.